CF1354C2 Not So Simple Polygon Embedding

The statement of this problem is the same as the statement of problem C1. The only difference is that, in problem C1, $ n $ is always even, and in C2, $ n $ is always odd. You are given a regular polygon with $ 2 \cdot n $ vertices (it's convex and has equal sides and equal angles) and all its sides have length $ 1 $ . Let's name it as $ 2n $ -gon. Your task is to find the square of the minimum size such that you can embed $ 2n $ -gon in the square. Embedding $ 2n $ -gon in the square means that you need to place $ 2n $ -gon in the square in such way that each point which lies inside or on a border of $ 2n $ -gon should also lie inside or on a border of the square. You can rotate $ 2n $ -gon and/or the square.

The first line contains a single integer $ T $ ( $ 1 \le T \le 200 $ ) — the number of test cases. Next $ T $ lines contain descriptions of test cases — one per line. Each line contains single odd integer $ n $ ( $ 3 \le n \le 199 $ ). Don't forget you need to embed $ 2n $ -gon, not an $ n $ -gon.

Print $ T $ real numbers — one per test case. For each test case, print the minimum length of a side of the square $ 2n $ -gon can be embedded in. Your answer will be considered correct if its absolute or relative error doesn't exceed $ 10^{-6} $ .